The primary goal of this chapter is to present a new method—called Logical Argument Mapping (LAM)—for the analysis of framing processes as they occur in any communication, but especially in conflicts. I start with a distinction between boundary setting, meaning construction, and sensemaking as three forms or aspects of framing, and argue that crucial for the resolution of frame-based controversies is our ability to deal with those “webs” of mutually supporting beliefs that determine sensemaking processes. Since any analysis of framing in conflicts and communication is itself influenced by sensemaking—there is no “frame-neutrality”—the main problem for an analyst is to cope with his or her own cognitive limitations. LAM offers a solution to this problem. The method will be exemplified with an analysis of two conflicting interpretations of how the international community should deal with Hamas after its election victory in 2006. (Note: Since in the published version mentioned below a section is missing, the pdf that you can download here is the only complete version of the paper.)